The E8 root system

Currently # roots fixed, # moved by π/3, # by π/2, # by 2π/3, # by π.
Actions: reinitialize, scramble, Weyl word simplify (# moves).
Show current Weyl word, set a word, apply a word. Show cycle structure.
Apply sample Weyl group elements: of order 8 (centralizer of order 192), of order 8 (centralizer of order 64), of order 30 (centralizer of order 30), of order 24 (centralizer of order 24), of order 20 (centralizer of order 20), of order 18 (centralizer of order 108), of order 14 (centralizer of order 28).
Choose projection: default, alternate, squares, Petrie (8 30-gons), 10 24-gons, 12 20-gons, 13 (blurred) 18-gons, 7 (blurred) 14-gons; randomly perturb projection.
Coloring: recolor to projection, toggle color scheme, briefly show displacement angles.

Mathematical background

[See also this page for a different interactive view of E8, showing its rotation under the Lie group G2.]

The display above shows a particular projection of the E8 root system, which can be described as a remarkable polytope in 8 dimensions (also known as the Gosset 421 polytope) having 240 vertices (known, in this context, as “roots”), and 6720 edges shown by black lines in the figure. Quite concretely, the roots can be described, in the coordinate system we have chosen, as the (112) points having coordinates (±1,±1,0,0,0,0,0,0) (where both signs can be chosen independently and the two non-zero coordinates can be anywhere) together with those (128) having coordinates (±½,±½,±½,±½,±½,±½,±½,±½) (where all signs can be chosen independently except that there must be an even number of minuses).

Every root (being identified with the vector leading from the origin to the vertex in question) is of the same length (i.e., all vertices are on a sphere with the origin as center; this is specific to E8 and is not a property of all root systems); the opposite of each root is again a root, and each one is orthogonal to 126 others, while forming an angle of π/3 with 56 others (those that are connected to it by an edge): the only possible angles between two roots are 0, π/3, π/2, 2π/3 and π.

The group of symmetries of this object is the group, known as the Weyl group of E8, generated by the (orthogonal) reflections about the hyperplane orthogonal to each root: this is a group of order 696729600 which can also be described as O8+(2). It is also the group of automorphism of the adjacency graph of the polytope.

Those 112 roots which have coordinates of the form (±1,±1,0,0,0,0,0,0) are shown as larger dots, and constitute a so-called D8 root system inside the E8 root system, which, as a polytope, is a rectified octacross; the reflections determined by those vertices generate a subgroup of order 5160960 (the Weyl group of D8, a subgroup of index 2 in {±1}≀𝕾8) of the full Weyl group of E8. The 128 remaining vertices (forming a demiocteract) are shown as smaller dots; alone, they are not a root system because the reflection determined by one of them does not fix that subset. Note that this division of the 240 vertices as 112+128 is particular to the chosen coordinate system and is not preserved by symmetries of the whole (except, precisely, by those living in the smaller Weyl group of D8; so there are 135 ways of making this decomposition).

One can further divide the roots in two by calling half of them “positive” in such a way that the sum of two positive roots, if it is a root, is always positive, and that for every root either it or its opposite is positive; there are many ways to do this (in fact, precisely as many as there are elements in the Weyl group), and we have chosen the division given by a lexicographic order on the coordinates: we call positive those roots such that the leftmost nonzero coordinate is positive (or, by numbering the roots lexicographically from 0 to 239, the positive ones are those numbered 120 through 239). A choice of positive roots is equivalent to a choice of fundamental (or simple) roots: these are the positive roots which cannot be written as a sum of two positive roots, and it then turns out that these form a basis of the ambient 8-space and, remarkably, that every positive root can be written as a linear combination of fundamental roots with nonnegative integer coefficients (equivalently, the fundamental roots form a non-orthogonal basis in which the coordinates of every root are either all nonnegative or all nonpositive; there is a uniquely defined greatest root, whose coordinates in terms of fundamental roots dominates that of every other root, and which happens to be one half the sum of all positive roots, fundamental or not: for E8, it is ⟨4,3,6,5,4,3,2,2⟩ and, for our choices, it is root number 239, or (1,1,0,0,0,0,0,0)). Any choice of positive/fundamental roots can be brought to any other choice by a unique element of the Weyl group.

If we represent the eight fundamental roots and connect two by a line whenever they form an angle of 2π/3 (the only other possibility being that they are orthogonal: in the case of E8, the angles of 3π/4 and 5π/6 do not occur), we obtain the so-called Dynkin diagram, which in the case of E8 has seven nodes in a simple chain and an eighth branching from the third. Here, we number the fundamental roots in the same total order as chosen to define the positive roots (i.e., lexicographic order on the coordinates; then the fundamental roots 1 through 8 are the roots numbered 120, 121, 122, 126, 132, 140, 150 and 162), and the Dynkin diagram has fundamental roots 8–1–3–4–5–6–7 in a chain and fundamental root number 2 branching off from 3.

The fundamental roots are important because the reflection with respect to them suffice to generate the Weyl group. Furthermore, the minimal length of an expression of a given element of the Weyl group as such a product of fundamental reflections (the length relative to the given element for the chosen system of fundamental roots) is equal to the number of positive roots whose image is a negative root; and composing by a fundamental reflection will always increase or decrease by 1 the length of the Weyl group element.

How to use this page

The graphic displays a particular two-dimensional projection of the eight-dimensional E8 root system. Various predefined projections are possible, emphasizing various symmetries in the Weyl group. The vertices (roots) are identified by their color and their size (vertices belonging to the D8 root system are shown larger); hovering the mouse cursor above a vertex shows information about the corresponding root, both the one which was initially present at this position and the one which has been placed their by applying whatever symmetry from the Weyl group: coordinates are given in the chosen orthonormal system and also as an integer (all nonnegative or all nonpositive) combination of fundamental roots, as well as the coordinates in the two-dimensional projection. The angle by which the root was displaced is also indicated.

Clicking on a root performs the reflection with respect to that root (it permutes that root with its opposite, fixes 126 others, and exchanges the 112 remaining roots as 56 pairs). The 696729600 elements of the Weyl group are generated by such reflections, and determine as many configurations of this widget. Reinitialize resets every root in its original position, whereas scramble chooses a random configuration (more or less uniformly in the Weyl group).

Each element of the Weyl group can be written as a product (of a uniquely defined length) of reflections by eight fundamental roots: the simplify command will perform a single step toward resolving the word by applying one of these fundamental reflections, the minimal number of which is displayed (the number of roots which are in their original place is also displayed). The show, set and apply commands are used to display the current Weyl group element as a word of minimal length in the letters 1 through 8 (indicating the fundamental reflections), set it to an arbitrary value, or compose it with an arbitrary value: thus, the simplify command essentially deletes the last letter in the word in question. Note that fundamental reflections mean relative to the original position of the fundamental roots, and are applied left to right in the word order; but it is equivalent to apply the reflections relative to their displaced position when reading the word right to left.

The show cycle structure command displays the number of cycles of each occurring length, of the current Weyl group element as acting on the 240 roots. The commands applying various sample Weyl group elements are given because (except the second element of order 8, which is a cyclic rotation of the coordinates in the chosen basis) they emphasize the symmetries of each of the predefined projections.

Most of the predefined projections have a rotational symmetry, meaning that the root projections are nicely arranged on regular n-gons: for example, one displays the 240 roots as 8 concentric 30-gons, the outer one being a so-called Petrie polygon for that polytope; there are, of course, a great many possible such projections, since one could apply any element of the Weyl group, but the one proposed here attempts to be very close to the default projection, so that passing from one projection to another will not move the vertices in too wild a fashion—however, this choice has not been made in a very systematic manner. The default (original) projection is related to the chosen coordinate system in that it can be described by linearly combining the coordinates with coefficients given by eight consecutive complex sixteenth roots of unity. There is also a command which applies a small random (gaussian) perturbation to the current projection, which gives a feel of how the root system can be moved slightly.

Note that while changing the projection moves the vertices and edges along, it does not change their color or size: the vertices remain colored as they were before the projection was changed. To apply a color scheme harmonious to the chosen projection (at least in the unscrambled state!), use the recolor to projection command. (This is typically what you want if you wish to visualize a given Weyl group elements in various projections; on the other hand, to compare the different projections with respect to each other, try changing projections without recoloring.) There is also a second color scheme (independent of projection) which can be used, in which positive roots (for the particular order chosen) are represented in blue and negative roots in green, and the eight fundamental roots (relative to that order) are labeled. Finally, the show displacement angles will temporarily replace the coloring of the roots do indicate by what angle they have been displaced under the current Weyl group element.